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Full text of "Treatise On Analysis Vol-Ii"

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(15.13.5) With the notation of (15.13.4), let EĢ be the orthogonal supple-
ment (dom(F))1 of the closed subspace F! =dom(f), and E^ the ortho-
gonal supplement (F(dom(F)))-L of the closed subspace F_i = F(F1).

(1 5.1 3.6) (i) The subspace EjJ (resp. E# ) is the set of all x e dom(#*) such that
H* - x = ix (resp. H* • x = — /x), anddom (//*) fa rte Jzrec/ JMWI 0/dom(#),

(ii) Lef G"1" (resp. G~) be the subspace of F(H*) H'Aoye ./fry* projection in
E is EĢ (resp. Ejp. Then G+ and G" are c/a*?d i /i E x E and T(H*) is the
Hilbert sum ofT(H)9 G+, and G~.

We have x e E# if and only if

for all j>edom(jfiO, or equivalently if and only if (H*y\x) = 0>|/x); by
definition (15.123), this signifies that x e dom(/f *) and H* • x = ix. Similarly
for x e E^ , if we observe that F(Fj) is the set of all H • y — z> with j; e dom(jy).
Let x be an arbitrary point in dom(H*); as E is the Hilbert sum of
dom(F) and E^, we may write H* -x+ix^H'y + iy + z with y e dom(H)
and z e E# . As we have H • y = H* - y and H* • z = iz, we may write
7 = H* • zt + &!, with zl = z/2z, hence H* • x + ix = H* • y + iy + H* • z^
+ /ZiJ this shows that z2 = ^c — y — zt satisfies H* • z2 = — iz2, in other
words z2 6 E^, and therefore domC/if*) is the sum of dom(jFf)ŧ E^, and E^.
This implies that F(H*) is the sum of F(H), G+ and G". That G+ (resp. G")
is closed follows from the fact that this subspace is the intersection of the
closed subspace F(H*) of E x E and of the subspace consisting of the points
(jc, y) such that y = ix (resp. y = — zx), which is obviously closed. It remains
to be shown that G4" , G", and T(H) are mutually orthogonal. The fact that
G+ and G~ are orthogonal follows from the relation

OX, ixi) | (x2 , - w:2)) = Oi I X2) + Oxi I -^2) = 0.

Furthermore, if x e dom(ff) and y e E#ŧ we have

((x,H-x)\(y9-iy)) = (x\y) + i(H-x\y) = (x\y) + &

and as H* • y = — zv, we see that G~ is orthogonal to F(H); the fact that
G+ is orthogonal to T(H) is proved in the same way, and this ends the proof
of (i) and (ii).ion (15.13.4) reduces the study of hermitian opera-